├── LICENSE ├── README.md ├── TEST3.m ├── angleAxis2Rot.m ├── cpMatrix.m ├── createLink.m ├── dhFwdKine.m ├── dhInvKine.m ├── dhTransform.m ├── newtonEuler.m ├── rot2AngleAxis.m ├── rot2RPY.m ├── rotX.m ├── rotY.m ├── rotZ.m ├── rpy2Rot.m ├── transform2Twist.m ├── twist2Transform.m └── velocityJacobian.m /LICENSE: -------------------------------------------------------------------------------- 1 | GNU GENERAL PUBLIC LICENSE 2 | Version 3, 29 June 2007 3 | 4 | Copyright (C) 2007 Free Software Foundation, Inc. 5 | Everyone is permitted to copy and distribute verbatim copies 6 | of this license document, but changing it is not allowed. 7 | 8 | Preamble 9 | 10 | The GNU General Public License is a free, copyleft license for 11 | software and other kinds of works. 12 | 13 | The licenses for most software and other practical works are designed 14 | to take away your freedom to share and change the works. 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If not, see . 649 | 650 | Also add information on how to contact you by electronic and paper mail. 651 | 652 | If the program does terminal interaction, make it output a short 653 | notice like this when it starts in an interactive mode: 654 | 655 | Copyright (C) 656 | This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'. 657 | This is free software, and you are welcome to redistribute it 658 | under certain conditions; type `show c' for details. 659 | 660 | The hypothetical commands `show w' and `show c' should show the appropriate 661 | parts of the General Public License. Of course, your program's commands 662 | might be different; for a GUI interface, you would use an "about box". 663 | 664 | You should also get your employer (if you work as a programmer) or school, 665 | if any, to sign a "copyright disclaimer" for the program, if necessary. 666 | For more information on this, and how to apply and follow the GNU GPL, see 667 | . 668 | 669 | The GNU General Public License does not permit incorporating your program 670 | into proprietary programs. If your program is a subroutine library, you 671 | may consider it more useful to permit linking proprietary applications with 672 | the library. If this is what you want to do, use the GNU Lesser General 673 | Public License instead of this License. But first, please read 674 | . 675 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # Robot-Arm-Kinematics-MATLAB-Function-package 2 | List of Usable functions in this package (More details documented in the MATLAB file comments) 3 | 4 | ### rotX 5 | - **R = rotX(theta):** 6 | 7 | Returns a rotation matrix describing a rotation about the X axis (theta in radians). 8 | 9 | 10 | ### rotY 11 | - **R = rotY(theta):** 12 | 13 | Returns a rotation matrix describing a rotation about the Y axis (theta in radians). 14 | 15 | 16 | ### rotZ 17 | - **R = rotZ(theta):** 18 | 19 | Returns a rotation matrix describing a rotation about the Z axis (theta in radians). 20 | 21 | 22 | ### rpy2Rot 23 | - **R = rpy2Rot (roll, pitch, yaw):** 24 | 25 | Returns a rotation matrix corresponding to a roll, pitch, yaw encoded rotation. 26 | 27 | 28 | ### rot2RPY 29 | - **[roll, pitch, yaw] = rot2RPY(R):** 30 | 31 | Returns the roll, pitch and yaw corresponding to a given rotation matrix. 32 | 33 | 34 | ### cpMatrix 35 | - **X = cpMatrix(w):** 36 | 37 | eturns the matrix packing of the cross product operator. I.E. Given vectors W and V, cpMatrix(W) * V = W x V 38 | 39 | 40 | ### angleAxis2Rot 41 | - **R = angleAxis2Rot(k, theta):** 42 | 43 | Returns the rotation matrix encoded by a rotation of theta radians about the unit vector k axis. 44 | 45 | 46 | ### rot2AngleAxis 47 | - **[k, theta] = rot2AngleAxis(R):** 48 | 49 | Returns the angle and axis corresponding to a rotation matrix. 50 | 51 | 52 | ### twist2Transform 53 | - **H = twist2Transform(t):** 54 | 55 | Returns the homogenous transformation matrix corresponding to a 6 element twist vector. 56 | 57 | 58 | ### transform2Twist 59 | - **t = transform2Twist(H):** 60 | 61 | Returns the twist vector corresponding to the provided homogenous transform matrix. 62 | 63 | 64 | ### dhTransform 65 | - **H = dhTransform(a, d, alpha, theta):** 66 | 67 | Returns the homogenous transform corresponding to the provide DH parameters for a link. 68 | 69 | 70 | ### createLink 71 | - **L = createLink(a, d, alpha, theta, centOfMass, mass, inertia):** 72 | 73 | Creates a **structure** with the following members: 74 | 75 | a – DH parameter a (meters) 76 | 77 | d – DH parameter d (meters) 78 | 79 | alpha – DH parameter alpha (radians) 80 | 81 | theta – DH parameter theta (radians) 82 | 83 | mass – link mass (kg) 84 | 85 | inertia – link mass moment of inertia (kg m^2) 86 | 87 | com – the position of the link’s center of mass 88 | 89 | isRotary – Boolean true if it is a rotary joint false if it is a prismatic joint. 90 | 91 | All vectors and tensors are to be expressed in the Link’s coordinate frame. 92 | 93 | 94 | ### dhFwdKine 95 | - **H = dhFwdKine(linkList, paramList):** 96 | 97 | Returns the forward kinematics of a manipulator with the provided DH parameter set. 98 | 99 | linkList is to be an array of links, each created by createLink 100 | 101 | paramList is to be an array containing the current state of their joint variables. 102 | 103 | 104 | ### velocityJacobian 105 | - **Jv = velocityJacobian(linkList):** 106 | 107 | Returns the velocity jacobian of the manipulator given an array of links created by the **createLink** function. 108 | 109 | 110 | ### dhInvKine 111 | - **[paramList, error] = dhInvKine (linkList, desTransform, paramListGuess):** 112 | 113 | Returns the parameter list necessary to achieve a desired homogenous transform and the residual error in that transform. 114 | 115 | linkList – a list of the joint parameters created by createLink 116 | 117 | 118 | ### newtonEuler 119 | - **[jointTorques, Jv, JvDot] = newtonEuler(linkList, paramList, paramListDot, paramListDDot, baseDynamics, endEffectorWrench, gravityDirection):** 120 | 121 | Computes the inverse dynamics of a serial link manipulator and provides the velocity jacobian and its rate of change. 122 | 123 | linkList – a list of the joint parameters created by createLink 124 | 125 | paramList – the current joint angles/distances 126 | 127 | paramListDot – the current joint angle/distance speeds 128 | 129 | paramListDDot – the current joint angle/distance accelerations 130 | 131 | baseDynamics – the angular velocity and acceleration of the base frame expressed in the base frame 132 | 133 | endEffectorWrench – the externally applied force and torque on the last frame expressed in the last frame. 134 | 135 | gravityDirection – the direction of gravity expressed in the base frame 136 | 137 | -------------------------------------------------------------------------------- /TEST3.m: -------------------------------------------------------------------------------- 1 | link1=createLink(0,0,0, [],[0;0;0] ,0, [0 0 0;0 0 0;0 0 0]); 2 | link2=createLink(0,0,pi/2,[],[0;0;0.125],5.325,[0.031 0 0;0 0.031 0;0 0 0.00666]); 3 | link3=createLink(1,0,0, [],[0.5;0;0] ,21.3, [1.788 0 0;0 1.788 0;0 0 0.027]); 4 | link4=createLink(0.5,0,0, 0 ,[0.25;0;0] ,10.65,[0.229 0 0;0 0.229 0;0 0 0.013]); 5 | linkList=[link1 link2 link3 link4]; 6 | 7 | paramList = [pi/2;2*pi/3;-pi/4;0]; 8 | paramListDot = [10;5;15;0]; 9 | paramListDDot = [2;-5;1;0]; 10 | 11 | baseDynamics.linA = [0;0;0]; 12 | baseDynamics.angV = [0;0;0]; 13 | baseDynamics.angA = [0;0;0]; 14 | endEffectorWrench = [0;0;0;0;0;0]; 15 | gravityDirection = [0;0;-1]; 16 | 17 | [jointTorques, Jv, JvDot] = newtonEuler( linkList,paramList, paramListDot, paramListDDot,baseDynamics, endEffectorWrench,gravityDirection ); 18 | disp(Jv) 19 | disp((JvDot)) 20 | disp(jointTorques) -------------------------------------------------------------------------------- /angleAxis2Rot.m: -------------------------------------------------------------------------------- 1 | % angleAxis2Rot - Returns the rotation matrix encoded by a rotation of theta 2 | % radians about the unit vector k axis. 3 | % 4 | % R = angleAxis2Rot(k, theta) 5 | % 6 | % by inputing the vector k as the rotation axis, and theta as the 7 | % rotation angle about the k axis, this function will return the 8 | % rotation matrix for this rotation. Using the unit quaternion 9 | % parameters. 10 | % 11 | % 12 | % R = the rotation matrix of the input rotation 13 | % k = the rotation axis in the form of 3x1 matrix 14 | % theta = the rotation angle about the k axis 15 | % 16 | % Michael Cheng 17 | % CWID: 10820067 18 | % MENG 544: Robot Mechanics: Kinematics, Dynamics, and Control 19 | % 9/29/2016 20 | 21 | function R = angleAxis2Rot(k, theta) 22 | e1 = k(1).*sin(theta./2); 23 | e2 = k(2).*sin(theta./2); 24 | e3 = k(3).*sin(theta./2); 25 | e4 = cos(theta./2); 26 | R = [1-(2.*((e2).^2))-(2.*((e3).^2)) 2.*(e1.*e2-e3.*e4) 2.*(e1.*e3+e2.*e4);2.*(e1.*e2+e3.*e4) 1-(2.*((e1).^2))-(2.*((e3).^2)) 2.*(e2.*e3-e1.*e4);2.*(e1.*e3-e2.*e4) 2.*(e2.*e3+e1.*e4) 1-(2.*((e1).^2))-(2.*((e2).^2))]; 27 | end -------------------------------------------------------------------------------- /cpMatrix.m: -------------------------------------------------------------------------------- 1 | % cpMatrix - Returns the matrix packing of the cross product operator. 2 | % 3 | % X = cpMatrix(w) 4 | % 5 | % By inputing a vector, this function will return a 3x3 matrix, this 6 | % matrix multiplied by another vector, will generate the cross 7 | % product of the two vectors. 8 | % 9 | % 10 | % X = the matrix packing the cross product operator of the input vector 11 | % w = the input vector in the form of a 3x1 matrix 12 | % 13 | % Michael Cheng 14 | % CWID: 10820067 15 | % MENG 544: Robot Mechanics: Kinematics, Dynamics, and Control 16 | % 9/29/2016 17 | 18 | function X = cpMatrix(w) 19 | X=[0 -w(3) w(2);w(3) 0 -w(1);-w(2) w(1) 0]; 20 | end -------------------------------------------------------------------------------- /createLink.m: -------------------------------------------------------------------------------- 1 | % createLink - Creates a structure with the DH parameters and some other 2 | % parameters. 3 | % 4 | % L = createLink(a, d, alpha, theta, centOfMass, mass, inertia, isRotary) 5 | % 6 | % With the input of DH parameters a, d, alpha and theta, and with the 7 | % input of the position of the center of mass, the mass of the link, 8 | % the inertia of the link, and the info of the type of link stored in 9 | % the isRotary parameter. 10 | % 11 | % L = the structure consisting the information of the link 12 | % a, alpha, d, theta = the DH parameters 13 | % centOfMass = the position of the center of mass 14 | % mass = the mass of the link 15 | % inertia = the inertia of the link 16 | % isRotary = determine whether the link is rotary or prismatic, true (1) 17 | % for rotary, false (2) for prismatic 18 | % 19 | % Michael Cheng 20 | % CWID: 10820067 21 | % MENG 544: Robot Mechanics: Kinematics, Dynamics, and Control 22 | % 9/29/2016 23 | 24 | function L = createLink(a, d, alpha, theta, centOfMass, mass, inertia) 25 | 26 | 27 | 28 | if isempty(theta)==1 29 | TF_isRotary=true; 30 | elseif isempty(d)==1 31 | TF_isRotary=false; 32 | else 33 | TF_isRotary=2; 34 | end 35 | 36 | L = struct('a', a, 'd', d, 'alpha', alpha, 'theta', theta, 'com', centOfMass, 'mass', mass, 'inertia', inertia, 'isRotary', TF_isRotary); 37 | end -------------------------------------------------------------------------------- /dhFwdKine.m: -------------------------------------------------------------------------------- 1 | % dhFwdKine1 - Returns the forward kinematics of a manipulator 2 | % with the provided DH parameter set. 3 | % 4 | % H = dhFwdKine1(linkList, paramList) 5 | % 6 | % With the input of the links consist in the array linkList, and the 7 | % current states of the joint variables consist in the array 8 | % paramList, using the DH transform, this function will return the 9 | % homogeneous transformation matrix. In this first version of the 10 | % forward kinematics function, the function checks the isRotary 11 | % parameter to determine if the link is rotary or prismatic. If the 12 | % link is rotary, the variable is for the theta parameter. If the 13 | % link is prismatic, the variable is for the d parameter. 14 | % 15 | % linkList = the array consisting all the link structures, every 16 | % structure consists all the information need for the link 17 | % paramList = the array that consists the variables of all the links 18 | % 19 | % Michael Cheng 20 | % CWID: 10820067 21 | % MENG 544: Robot Mechanics: Kinematics, Dynamics, and Control 22 | % 9/29/2016 23 | 24 | function H = dhFwdKine(linkList, paramList) 25 | A = length(linkList); 26 | %syms n m; 27 | z=1; 28 | for n = 1:1:A 29 | 30 | if linkList(n).isRotary == 1 31 | linkList(n).theta=paramList(z); 32 | z=z+1; 33 | elseif linkList(n).isRotary == 0 34 | linkList(n).d = paramList(z); 35 | z=z+1; 36 | else 37 | 38 | end 39 | end 40 | H = 1; 41 | for m = 1:1:A 42 | H = H*dhTransform(linkList(m).a,linkList(m).d,linkList(m).alpha,linkList(m).theta); 43 | end 44 | end -------------------------------------------------------------------------------- /dhInvKine.m: -------------------------------------------------------------------------------- 1 | % dhInvKine - Returns the parameter list necessary to achieve a desired 2 | % homogenous transform and the residual error in that transform. 3 | % 4 | % [paramList, error] = dhInvKine (linkList, desTransform, paramListGuess) 5 | % 6 | % With the input of the links consist in the array linkList, the 7 | % desired transformation matrix which represents the desired tool position, 8 | % and an initial parameter guess, which will be corrected closer and 9 | % closer to the desired values throughout the function loop. The 10 | % function will return the final joint values and the residual error. 11 | % 12 | % linkList = the array consisting all the link structures, every 13 | % structure consists all the information need for the link 14 | % desTransform = the desired transformation matrix of the desired tool 15 | % position 16 | % paramListGuess = an initial guess of where the joint variables 17 | % currently are, could be the joints current position 18 | % paramList = returns the desired joint variable positions 19 | % error = the residual error form the function loop 20 | % 21 | % Michael Cheng 22 | % CWID: 10820067 23 | % MENG 544: Robot Mechanics: Kinematics, Dynamics, and Control 24 | % 11/13/2016 25 | 26 | function [paramList, error] = dhInvKine (linkList, desTransform, paramListGuess) 27 | 28 | %desired 29 | Rd = desTransform(1:3,1:3); 30 | Pd = desTransform(1:3,4); 31 | theta_d = acos((Rd(1,1)+Rd(2,2)+Rd(3,3)-1)./2); 32 | k_d = (1./(2.*sin(theta_d)))*[Rd(3,2)-Rd(2,3);Rd(1,3)-Rd(3,1);Rd(2,1)-Rd(1,2)]; 33 | q_d1 = k_d(1).*sin(theta_d./2); 34 | q_d2 = k_d(2).*sin(theta_d./2); 35 | q_d3 = k_d(3).*sin(theta_d./2); 36 | q_d0 = cos(theta_d./2); 37 | Td = [Pd;q_d0;q_d1;q_d2;q_d3]; 38 | 39 | %check which link has no variables, ensure which joint variables don't change 40 | A=length(linkList); 41 | z=1; 42 | for G=1:1:A 43 | if linkList(G).isRotary==2 44 | Y(z)=G; 45 | z=z+1; 46 | end 47 | end 48 | 49 | %current 50 | q_current = paramListGuess; 51 | go=true; 52 | while go 53 | curTransform = dhFwdKine(linkList, q_current); 54 | Rc = curTransform(1:3,1:3); 55 | Pc = curTransform(1:3,4); 56 | theta_c = acos((Rc(1,1)+Rc(2,2)+Rc(3,3)-1)./2); 57 | k_c = (1./(2.*sin(theta_c)))*[Rc(3,2)-Rc(2,3);Rc(1,3)-Rc(3,1);Rc(2,1)-Rc(1,2)]; 58 | q_c1 = k_c(1).*sin(theta_c./2); 59 | q_c2 = k_c(2).*sin(theta_c./2); 60 | q_c3 = k_c(3).*sin(theta_c./2); 61 | q_c0 = cos(theta_c./2); 62 | Tc = [Pc;q_c0;q_c1;q_c2;q_c3]; 63 | C = (1/2)*[-q_c1 -q_c2 -q_c3;q_c0 -q_c3 q_c2;q_c3 q_c0 -q_c1;-q_c2 q_c1 q_c0]; 64 | Jvc = velocityJacobian( linkList, q_current ); 65 | Jg = [eye(3) zeros(3);zeros(4,3) C]*Jvc; 66 | %pseudo inverse 67 | [U,S,V] = svd(Jg); 68 | %[m,n]=size(Jg); 69 | %r=rank(S); 70 | %SR=S(1:r,1:r); 71 | %SRc=[SR^-1 zeros(r,m-r);zeros(n-r,r) zeros(n-r,m-r)]; 72 | %Jg_pseuInv=V*SRc*U.'; 73 | Jg_pseuInv=V*pinv(S)*U'; 74 | e=Td-Tc; 75 | q_delta=-Jg_pseuInv*e; 76 | q_current=q_current-q_delta; 77 | %the loop stops when all values in delta q is smaller than 0.001 78 | if (abs(q_delta(:))<=0.001) 79 | go = false; 80 | end 81 | end 82 | 83 | %joints that aren't rotary don't change, so those joint values remains same 84 | %as the paramList 85 | q_delta(Y)=0; 86 | q_current(Y)=paramListGuess(Y); 87 | paramList = q_current; 88 | error = q_delta; 89 | 90 | paramList = double(paramList); 91 | error = double(error); 92 | -------------------------------------------------------------------------------- /dhTransform.m: -------------------------------------------------------------------------------- 1 | % dhTransform - Returns the homogenous transform corresponding 2 | % to the provide DH parameters for a link. 3 | % 4 | % H = dhTransform(a, d, alpha, theta) 5 | % 6 | % With the input of DH parameters a, d, alpha and theta, this 7 | % function will return the homogeneous transformation matrix, which 8 | % is the product of transX(a), rotX(alpha), transZ(d), rotZ(theta). 9 | % 10 | % a, alpha, d, theta = the DH parameters 11 | % H = the 4x4 homogeneous transformation matrix corresponding to the DH 12 | % parameters 13 | % 14 | % Michael Cheng 15 | % CWID: 10820067 16 | % MENG 544: Robot Mechanics: Kinematics, Dynamics, and Control 17 | % 9/29/2016 18 | 19 | function H = dhTransform(a, d, alpha, theta) 20 | H = [1 0 0 a;0 1 0 0;0 0 1 0;0 0 0 1] * [rotX(alpha) [0;0;0];0 0 0 1] * [1 0 0 0;0 1 0 0;0 0 1 d;0 0 0 1] * [rotZ(theta) [0;0;0];0 0 0 1]; 21 | end -------------------------------------------------------------------------------- /newtonEuler.m: -------------------------------------------------------------------------------- 1 | % newtonEuler - Computes the inverse dynamics of a serial link manipulator 2 | % and provides the velocity jacobian and its rate of change. 3 | % 4 | % [jointTorques, Jv, JvDot] = newtonEuler(linkList, paramList, paramListDot, paramListDDot,baseDynamics, endEffectorWrench,gravityDirection) 5 | % 6 | % With the input of the links consist in the array linkList, the 7 | % current states of the joint variables consist in the array 8 | % paramList, the current rate of change of joint variables in the 9 | % array paramListDot, and the rate of chenge of that in 10 | % paramListDDot, the base dynamics of the machanical structure, the 11 | % force and torque applied on the end effector of the machanical 12 | % structure, and the gravitational pull direction, this function 13 | % returns the torque of each joint, the velocity jacobian, and the 14 | % time derivative of the jacobian. 15 | % 16 | % linkList = the array consisting all the link structures, every 17 | % structure consists all the information need for the link 18 | % paramList = the array that consists the current joint variable 19 | % positions 20 | % paramListDot = the array that consists the rate of change of the joint 21 | % variables 22 | % baseDynamics = the the angular velocity and acceleration of the base 23 | % frame expressed in the base frame 24 | % endEffectorWrench = the externally applied force and torque on the last 25 | % frame expressed in the last frame. 26 | % gravityDirection = the direction of gravity expressed in the base frame 27 | % jointTorques = returns the torque of each joint 28 | % Jv = returns the velocity jacobian of the system 29 | % JvDot = returns the time derivative of the velocity jacobian 30 | % 31 | % Michael Cheng 32 | % CWID: 10820067 33 | % MENG 544: Robot Mechanics: Kinematics, Dynamics, and Control 34 | % 11/13/2016 35 | 36 | function [jointTorques, Jv, JvDot] = newtonEuler( linkList,paramList, paramListDot, paramListDDot,baseDynamics, endEffectorWrench,gravityDirection ) 37 | 38 | %Prellocating arrays 39 | 40 | linkListSym = repmat(linkList,1); 41 | 42 | A=length(linkList); 43 | Pc = zeros(3,1,A); 44 | m = zeros(A,1); 45 | I = zeros(3,3,A); 46 | FwdLinkTransforms = zeros(4,4,A); 47 | InvLinkTransforms = zeros(4,4,A); 48 | FwdR = zeros(3,3,A); 49 | InvR = zeros(3,3,A); 50 | FwdP = zeros(3,1,A); 51 | InvP = zeros(3,1,A); 52 | w = zeros(3,1,A+1); 53 | w_dot = zeros(3,1,A+1); 54 | v_dot = zeros(3,1,A+1); 55 | vc_dot = zeros(3,1,A); 56 | F = zeros(3,1,A); 57 | N = zeros(3,1,A); 58 | f = zeros(3,1,A+1); 59 | n_inward = zeros(3,1,A+1); 60 | torque = zeros(A,1); 61 | 62 | %Inputing the base dynamics 63 | w(:,:,1) = baseDynamics.angV; 64 | w_dot(:,:,1) = baseDynamics.angA; 65 | v_dot(:,:,1) = baseDynamics.linA + 9.8*gravityDirection; 66 | 67 | %Putting paramList values in to linkList 68 | z=1; 69 | for n = 1:1:A 70 | if linkList(n).isRotary == 1 71 | linkList(n).theta=paramList(z); 72 | z=z+1; 73 | elseif linkList(n).isRotary == 0 74 | linkList(n).d = paramList(z); 75 | z=z+1; 76 | else 77 | end 78 | end 79 | 80 | %Taking the centOfMass from each link and putting them into a list: Pc 81 | for n = 1:1:A 82 | Pc(:,:,n) = linkList(n).com; 83 | end 84 | 85 | %Taking the mass from each link and putting them into a list: m 86 | for n = 1:1:A 87 | m(n) = linkList(n).mass; 88 | end 89 | 90 | %Taking the inertia from each link and putting them into a list: I 91 | for n = 1:1:A 92 | I(:,:,n) = linkList(n).inertia; 93 | end 94 | 95 | %Getting forward transform list: FwdLinkTransforms 96 | for n = 1:1:A 97 | FwdLinkTransforms(:,:,n) = dhTransform(linkList(n).a,linkList(n).d,linkList(n).alpha,linkList(n).theta); 98 | end 99 | 100 | %Getting inverse transform list: InvLinkTransforms 101 | for n = 1:1:A 102 | InvLinkTransforms(:,:,n) = inv(FwdLinkTransforms(:,:,n)); 103 | end 104 | 105 | %Getting forward rotation matrix list: FwdR 106 | for n = 1:1:A 107 | FwdR(:,:,n) = FwdLinkTransforms(1:3,1:3,n); 108 | end 109 | 110 | %Getting inverse rotation matrix list: InvR 111 | for n = 1:1:A 112 | InvR(:,:,n) = InvLinkTransforms(1:3,1:3,n); 113 | end 114 | 115 | %Getting forward position matrix list: FwdP 116 | for n = 1:1:A 117 | FwdP(:,:,n) = FwdLinkTransforms(1:3,4,n); 118 | end 119 | 120 | %Getting inverse position matrix list: InvP 121 | for n = 1:1:A 122 | InvP(:,:,n) = InvLinkTransforms(1:3,4,n); 123 | end 124 | 125 | %find w list: w 126 | for i = 0:1:A-1 127 | w(:,:,(i+1)+1) = InvR(:,:,(i)+1)*w(:,:,(i)+1) + paramListDot(i+1)*[0;0;1]; 128 | end 129 | 130 | %find w_dot list: w_dot 131 | for i = 0:1:A-1 132 | w_dot(:,:,(i+1)+1) = InvR(:,:,(i)+1)*w_dot(:,:,(i)+1) + cpMatrix(InvR(:,:,(i)+1)*w(:,:,(i)+1))*paramListDot(i+1)*[0;0;1] + paramListDDot(i+1)*[0;0;1]; 133 | end 134 | 135 | %find v_dot list: v_dot 136 | for i = 0:1:A-1 137 | v_dot(:,:,(i+1)+1) = InvR(:,:,(i)+1)*(cross(w_dot(:,:,(i)+1),FwdP(:,:,(i)+1)) + cpMatrix(w(:,:,(i)+1))*cpMatrix(w(:,:,(i)+1))*FwdP(:,:,(i)+1) + v_dot(:,:,(i)+1)); 138 | end 139 | 140 | %find vc_dot list: vc_dot 141 | for i = 0:1:A-1 142 | vc_dot(:,:,(i)+1) = cross(w_dot(:,:,(i+1)+1),Pc(:,:,i+1)) + cpMatrix(w(:,:,(i+1)+1))*cpMatrix(w(:,:,(i+1)+1))*Pc(:,:,(i)+1) + v_dot(:,:,(i+1)+1); 143 | end 144 | 145 | %find F list: F 146 | for i = 0:1:A-1 147 | F(:,:,(i)+1) = m((i)+1)*vc_dot(:,:,(i)+1); 148 | end 149 | 150 | %find N list: N 151 | for i = 0:1:A-1 152 | N(:,:,(i)+1) = I(:,:,(i)+1)*w_dot(:,:,(i+1)+1) + cpMatrix(w(:,:,(i+1)+1))*I(:,:,(i)+1)*w(:,:,(i+1)+1); 153 | end 154 | 155 | %find f list: f 156 | for i = A:-1:1 157 | f(:,:,i) = FwdR(:,:,i)*f(:,:,i+1) + F(:,:,i); 158 | end 159 | 160 | %find n list: n 161 | for i = A:-1:1 162 | n_inward(:,:,i) = N(:,:,i) + FwdR(:,:,i)*n_inward(:,:,i+1) + cpMatrix(Pc(:,:,i))*F(:,:,i) + cpMatrix(FwdP(:,:,i))*FwdR(:,:,i)*f(:,:,i+1); 163 | end 164 | 165 | %find torque list: torque 166 | for i = 1:1:A 167 | torque(i) = n_inward(:,:,i)'*[0;0;1]; 168 | end 169 | 170 | %Calculate Jv 171 | Jv = (velocityJacobian(linkList, paramList)); 172 | 173 | %Calculate the joint torques 174 | t_endEffector = Jv'*endEffectorWrench; 175 | jointTorques = torque + t_endEffector; 176 | 177 | 178 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 179 | %Compute JvDot 180 | 181 | %Setting up the symbollic variables 182 | thetaVars = sym('theta', [A 1]); 183 | dVars = sym('d', [A 1]); 184 | thetaDotVars = sym('thetaDot', [A 1]); 185 | dDotVars = sym('dDot', [A 1]); 186 | thetaCount=0; 187 | dCount=0; 188 | %Count the quantity of the variables in the linkList 189 | for i = 1:1:A 190 | if linkListSym(i).isRotary==1 191 | thetaCount=thetaCount+1; 192 | elseif linkListSym(i).isRotary==0 193 | dCount=dCount+1; 194 | end 195 | end 196 | %Putting the symbollic variables in an array in the form of paramList: var 197 | varCur = 1; 198 | numOfVar = thetaCount + dCount; 199 | var=rand(numOfVar,1); 200 | for i = 1:1:A 201 | if linkListSym(i).isRotary==1 202 | var=subs(var,var(varCur),thetaVars(i)); 203 | varCur = varCur+1; 204 | end 205 | if linkListSym(i).isRotary==0 206 | var=subs(var,var(varCur),dVars(i)); 207 | varCur = varCur+1; 208 | end 209 | end 210 | %Putting the time derivative of the symbollic variables in an array in the 211 | %form of paramListDot: varDot 212 | varDotCur = 1; 213 | varDot=rand(numOfVar,1); 214 | for i = 1:1:A 215 | if linkListSym(i).isRotary==1 216 | varDot=subs(varDot,varDot(varDotCur),thetaDotVars(i)); 217 | varDotCur = varDotCur+1; 218 | end 219 | if linkListSym(i).isRotary==0 220 | varDot=subs(varDot,varDot(varDotCur),dDotVars(i)); 221 | varDotCur = varDotCur+1; 222 | end 223 | end 224 | %Construct an symbollic representation of the velocity jacobian: Jv_symbol 225 | Jv_symbol = (velocityJacobian(linkListSym,var)); 226 | %Construct symbollic JvDot 227 | B = length(var); 228 | JvDot = 0; 229 | for i = 1:1:B 230 | JvDot = JvDot + diff(Jv_symbol,var(i))*varDot(i); 231 | end 232 | %Substitute symbollic variables with actual corresponding paramList values 233 | for i = 1:1:B 234 | JvDot = subs(JvDot,var(i),paramList(i)); 235 | end 236 | for i = 1:1:B 237 | JvDot = subs(JvDot,varDot(i),paramListDot(i)); 238 | end 239 | 240 | Jv = double(Jv); 241 | JvDot = double(JvDot); 242 | jointTorques = double(jointTorques); -------------------------------------------------------------------------------- /rot2AngleAxis.m: -------------------------------------------------------------------------------- 1 | % rot2AngleAxis - Returns the angle and axis corresponding to a 2 | % rotation matrix. 3 | % 4 | % [k, theta] = rot2AngleAxis(R) 5 | % 6 | % by inputing the rotation matrix, the function will return the 7 | % array, consisting the corresponding rotation axis k, and rotation 8 | % angle theta 9 | % 10 | % 11 | % k = the rotation axis in the form of 3x1 matrix 12 | % theta = the rotation angle about the k axis 13 | % R = the input rotation matrix 14 | % 15 | % Michael Cheng 16 | % CWID: 10820067 17 | % MENG 544: Robot Mechanics: Kinematics, Dynamics, and Control 18 | % 9/29/2016 19 | 20 | function [k, theta] = rot2AngleAxis(R) 21 | 22 | theta = acos((R(1,1)+R(2,2)+R(3,3)-1)./2); 23 | k = (1./(2.*sin(theta)))*[R(3,2)-R(2,3);R(1,3)-R(3,1);R(2,1)-R(1,2)]; 24 | 25 | end -------------------------------------------------------------------------------- /rot2RPY.m: -------------------------------------------------------------------------------- 1 | % rot2RPY - Generates the roll, pitch, yaw angles from a rotational transformation matrix. 2 | % 3 | % [roll, pitch, yaw] = rot2RPY(R) - 4 | % 5 | % By inputing the rotation matrix, this function will generate the 6 | % roll, pitch, yaw angles of this rotation matrix in an output array. 7 | % 8 | % roll = the angle of rotation about the X axis, in radians 9 | % pitch = the angle of rotation about the Y axis, in radians 10 | % yaw = the angle of rotation about the Z axis, in radians 11 | % R = the input 3x3 rotation matrix 12 | % 13 | % Michael Cheng 14 | % CWID: 10820067 15 | % MENG 544: Robot Mechanics: Kinematics, Dynamics, and Control 16 | % 9/29/2016 17 | 18 | function [roll, pitch, yaw] = rot2RPY(R) 19 | pitch = atan2(-R(3,1), (((R(1,1)).^2) + ((R(2,1)).^2)).^(1/2)); 20 | yaw = atan2(R(2,1)./cos(pitch), R(1,1)./cos(pitch)); 21 | roll = atan2(R(3,2)./cos(pitch), R(3,3)./cos(pitch)); 22 | end -------------------------------------------------------------------------------- /rotX.m: -------------------------------------------------------------------------------- 1 | % rotX - Generates a rotation matrix rotating about the X axis by theta. 2 | % 3 | % R = rotX(theta) - 4 | % 5 | % By inputing a theta, in radians, this function will generate a 6 | % 3x3 rotation matrix. When a vector is multiplied by this rotation 7 | % matrix R, the vector will rotate about the X axis by theta. 8 | % 9 | % R = the 3x3 rotation matrix about the X axis by theta 10 | % theta = the angle of rotation about the X axis, in radians 11 | % 12 | % Michael Cheng 13 | % CWID: 10820067 14 | % MENG 544: Robot Mechanics: Kinematics, Dynamics, and Control 15 | % 9/29/2016 16 | 17 | function R = rotX(theta) 18 | R=[1 0 0;0 cos(theta) -sin(theta);0 sin(theta) cos(theta)]; 19 | end -------------------------------------------------------------------------------- /rotY.m: -------------------------------------------------------------------------------- 1 | % rotY - Generates a rotation matrix rotating about the Y axis by theta. 2 | % 3 | % R = rotY(theta) - 4 | % 5 | % By inputing a theta, in radians, this function will generate a 6 | % 3x3 rotation matrix. When a vector is multiplied by this rotation 7 | % matrix R, the vector will rotate about the Y axis by theta. 8 | % 9 | % R = the 3x3 rotation matrix about the Y axis by theta 10 | % theta = the angle of rotation about the Y axis, in radians 11 | % 12 | % Michael Cheng 13 | % CWID: 10820067 14 | % MENG 544: Robot Mechanics: Kinematics, Dynamics, and Control 15 | % 9/29/2016 16 | 17 | function R = rotY(theta) 18 | R=[cos(theta) 0 sin(theta);0 1 0;-sin(theta) 0 cos(theta)]; 19 | end -------------------------------------------------------------------------------- /rotZ.m: -------------------------------------------------------------------------------- 1 | % rotZ - Generates a rotation matrix rotating about the Z axis by theta. 2 | % 3 | % R = rotZ(theta) - 4 | % 5 | % By inputing a theta, in radians, this function will generate a 6 | % 3x3 rotation matrix. When a vector is multiplied by this rotation 7 | % matrix R, the vector will rotate about the Z axis by theta. 8 | % 9 | % R = the 3x3 rotation matrix about the Z axis by theta 10 | % theta = the angle of rotation about the Z axis, in radians 11 | % 12 | % Michael Cheng 13 | % CWID: 10820067 14 | % MENG 544: Robot Mechanics: Kinematics, Dynamics, and Control 15 | % 9/29/2016 16 | 17 | function R = rotZ(theta) 18 | R=[cos(theta) -sin(theta) 0;sin(theta) cos(theta) 0;0 0 1]; 19 | end -------------------------------------------------------------------------------- /rpy2Rot.m: -------------------------------------------------------------------------------- 1 | % rpy2Rot - Generates a rotational transformation matrix from roll, pitch, yaw angles. 2 | % 3 | % R = rpy2Rot(roll, pitch, yaw) - 4 | % 5 | % By inputing roll, pitch, yaw angles, in radians, this function will 6 | % generate a transformation matrix. When a vector is multiplied by this 7 | % rotation matrix R, the vector will rotate about X, Y, Z axis by the 8 | % angle of roll, pitch, yaw. 9 | % 10 | % R = the 3x3 rotational transformation matrix after roll, pitch, yaw rotations 11 | % roll = the angle of rotation about the X axis, in radians 12 | % pitch = the angle of rotation about the Y axis, in radians 13 | % yaw = the angle of rotation about the Z axis, in radians 14 | % 15 | % Michael Cheng 16 | % CWID: 10820067 17 | % MENG 544: Robot Mechanics: Kinematics, Dynamics, and Control 18 | % 9/29/2016 19 | 20 | function R = rpy2Rot (roll, pitch, yaw) 21 | R=[cos(yaw) -sin(yaw) 0;sin(yaw) cos(yaw) 0;0 0 1]*[cos(pitch) 0 sin(pitch);0 1 0;-sin(pitch) 0 cos(pitch)]*[1 0 0;0 cos(roll) -sin(roll);0 sin(roll) cos(roll)]; 22 | end -------------------------------------------------------------------------------- /transform2Twist.m: -------------------------------------------------------------------------------- 1 | % transform2Twist - Returns the twist vector corresponding to the 2 | % provided homogenous transform matrix. 3 | % 4 | % t = transform2Twist(H) 5 | % 6 | % With the input of a homogeneous transformation matrix, the function 7 | % returns the twist function correspondingly, in the form of a 6x1 matrix 8 | % with the first 3 elements being the v vector and the last three elements 9 | % being the w vector. 10 | % 11 | % t = the 6 element input twist vector in the form of 6x1 matrix 12 | % H = the 4x4 homogeneous transformation matrix 13 | % 14 | % Michael Cheng 15 | % CWID: 10820067 16 | % MENG 544: Robot Mechanics: Kinematics, Dynamics, and Control 17 | % 9/29/2016 18 | 19 | function t = transform2Twist(H) 20 | R = [H(1,1) H(1,2) H(1,3);H(2,1) H(2,2) H(2,3);H(3,1) H(3,2) H(3,3)]; 21 | theta = acos((trace(R)-1)./2); 22 | Wunit = (1./(2.*sin(theta)))*[R(3,2)-R(2,3);R(1,3)-R(3,1);R(2,1)-R(1,2)]; 23 | w = Wunit.*theta; 24 | Wunitcp = [0 -Wunit(3) Wunit(2);Wunit(3) 0 -Wunit(1);-Wunit(2) Wunit(1) 0]; 25 | v = inv(((eye(3)-R)*Wunitcp)+(theta*w*transpose(w)))*[H(1,4);H(2,4);H(3,4)]; 26 | t = [v;w]; 27 | end -------------------------------------------------------------------------------- /twist2Transform.m: -------------------------------------------------------------------------------- 1 | % twist2Transform - Returns the homogenous transformation matrix 2 | % corresponding to a 6 element twist vector. 3 | % 4 | % H = twist2Transform(t) 5 | % 6 | % With the twist vector in the form of 6x1 matrix, which consists the 7 | % w and the v vectors, this function calculates theta, the unit w 8 | % vector, and the matrix packing of the cross product operator of unit 9 | % w vector. 10 | % 11 | % H = the homogeneous transformation matrix for the twist vector 12 | % t = the 6 element twist vector in the form of 6x1 matrix 13 | % 14 | % Michael Cheng 15 | % CWID: 10820067 16 | % MENG 544: Robot Mechanics: Kinematics, Dynamics, and Control 17 | % 9/29/2016 18 | 19 | function H = twist2Transform(t) 20 | v = [t(1);t(2);t(3)]; 21 | w = [t(4);t(5);t(6)]; 22 | theta = (((t(4)).^2)+((t(5)).^2)+((t(6)).^2)).^(1/2); 23 | Wunit = w./theta; 24 | Wunitcp = [0 -Wunit(3) Wunit(2);Wunit(3) 0 -Wunit(1);-Wunit(2) Wunit(1) 0]; 25 | e = (cos(theta)*eye(3))+(sin(theta)*Wunitcp)+((1-cos(theta))*Wunit*transpose(Wunit)); 26 | d = (((eye(3)-e)*Wunitcp)+(theta*Wunit*transpose(Wunit)))*v; 27 | g = [0 0 0 1]; 28 | H = [e d;g]; 29 | end 30 | 31 | 32 | -------------------------------------------------------------------------------- /velocityJacobian.m: -------------------------------------------------------------------------------- 1 | % velocityJacobian - Returns the velocity jacobian of the manipulator given 2 | % an array of links created by the createLink function and 3 | % the current joint variables. 4 | % 5 | % Jv = velocityJacobian( linkList, paramList ) 6 | % 7 | % By inputting the link list and the current state parameter list, 8 | % this function returns the velocity jacobian of this set of links. 9 | % 10 | % linkList = the array consisting all the link structures, every 11 | % structure consists all the information need for the link 12 | % paramList = the array that consists the joint variables 13 | % 14 | % Michael Cheng 15 | % CWID: 10820067 16 | % MENG 544: Robot Mechanics: Kinematics, Dynamics, and Control 17 | % 11/13/2016 18 | function Jv = velocityJacobian( linkList, paramList ) 19 | 20 | H=dhFwdKine(linkList,paramList); 21 | A=length(linkList); 22 | 23 | %Putting paramList values in to linkList 24 | z=1; 25 | for n = 1:1:A 26 | if linkList(n).isRotary == 1 27 | linkList(n).theta=paramList(z); 28 | z=z+1; 29 | elseif linkList(n).isRotary == 0 30 | linkList(n).d = paramList(z); 31 | z=z+1; 32 | else 33 | end 34 | end 35 | 36 | R0_N=H(1:3,1:3); 37 | TN_0=inv(H); 38 | 39 | %Prellocating arrays 40 | TN_i=sym(zeros(4,4,A)); 41 | z=sym(zeros(3,1,A)); 42 | d=sym(zeros(3,1,A)); 43 | JvN=sym(zeros(6,A)); 44 | 45 | %Getting the list of TN_i 46 | T_cur=TN_0; 47 | for n=1:1:A 48 | T_cur=T_cur*dhTransform(linkList(n).a,linkList(n).d,linkList(n).alpha,linkList(n).theta); 49 | TN_i(:,:,n)=T_cur; 50 | end 51 | 52 | %Getting the list of z from TN_i 53 | for m=1:1:A 54 | z(:,:,m)=TN_i(1:3,1:3,m)*[0;0;1]; 55 | end 56 | 57 | %Getting the list of d from TN_i 58 | for p=1:1:A 59 | d(:,:,p)=-TN_i(1:3,4,p); 60 | end 61 | 62 | %Construct JvN with z and d 63 | for q=1:1:A 64 | JvN(:,q)=[cross(z(:,:,q),d(:,:,q));z(:,:,q)]; 65 | end 66 | 67 | %Get Jv from JvN 68 | Jv=[R0_N zeros(3,3);zeros(3,3) R0_N]*JvN; 69 | 70 | --------------------------------------------------------------------------------